\def\g{{\frak g}}
\def\C{{\Bbb C}}
\def\K{{\Bbb K}}
\def\R{{\Bbb R}}
\def\Aut{\mathop{\rm Aut}\nolimits}
\def\sdir#1{\hbox{$\mathrel\times{\hskip -4.6pt
{\vrule height 4.7 pt depth .5 pt}}\hskip 2pt_{#1}$}}
Let $\g$ be a simple Lie algebra of finite dimension over $\K
\in \left\{\R,\C\right\}$ and $\Aut(\g)$ the finite-dimensional
Lie group of its automorphisms. We will calculate the component
group $\pi_0(\Aut(\g)) = \Aut(\g)/\Aut(\g)_0$ and the number of
its conjugacy classes, and we will show that the corresponding
short exact sequence
$$
{\bf1}\to\Aut(\g)_0\to\Aut(\g)\to\pi_0(\Aut(\g))\to{\bf1}
$$
is split or, equivalently, there is an isomorphism $\Aut(\g)\cong
\Aut(\g)_0 \sdir{}\pi_0(\Aut(\g))$. Indeed, since $\Aut(\g)_0$ is
open in $\Aut(\g)$, the quotient group $\pi_0(\Aut(\g))$ is discrete.
Hence a section $\pi_0(\Aut(\g))\to\Aut(\g)$ is automatically continuous,
giving rise to an isomorphism of Lie groups $\Aut(\g)\cong\Aut(\g)_0
\sdir{}\pi_0(\Aut(\g))$.